Mehler kernel#Fractional Fourier transform

{{harvs|txt|last=Mehler|authorlink=Gustav Ferdinand Mehler|year=1866}} defined a function{{Citation | last1=Mehler | first1=F. G. | title=Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung | url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002152975 | language=German |id={{ERAM|066.1720cj}} | year=1866 | journal=Journal für die Reine und Angewandte Mathematik | issn=0075-4102 | issue=66 | pages=161–176}} (cf. p 174, eqn (18) & p 173, eqn (13) )

{{Equation box 1

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|equation = $E(x,y)\; =\backslash frac\; 1\{\backslash sqrt\{1-\backslash rho^2\}\}\backslash exp\backslash left(-\backslash frac\{\backslash rho^2\; (x^2+y^2)-\; 2\backslash rho\; xy\}\{(1-\backslash rho^2)\}\backslash right)~,$

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and showed, in modernized notation,{{citation|first1=Arthur|last1= Erdélyi|authorlink1=Arthur Erdélyi|first2= Wilhelm |last2=Magnus|authorlink2=Wilhelm Magnus|first3= Fritz|last3= Oberhettinger|first4=Francesco G.|last4= Tricomi|authorlink4=Francesco Tricomi|title= Higher transcendental functions. Vol. II|publisher= McGraw-Hill| year=1955}} ([http://www.nr.com/legacybooks scan]:   [http://apps.nrbook.com/bateman/Vol2.pdf p.194 10.13 (22)]) that it can be expanded in terms of Hermite polynomials $H(\backslash cdot)$ based on weight function $\backslash exp(-x^2)$ as

:$E(x,y)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(\backslash rho/2)^n\}\{n!\}\; ~\; \backslash mathit\{H\}\_n(x)\backslash mathit\{H\}\_n(y)\; ~.$

This result is useful, in modified form, in quantum physics, probability theory, and harmonic analysis.

In physics, the fundamental solution, (Green's function), or propagator of the Hamiltonian for the quantum harmonic oscillator is called the Mehler kernel. It provides the fundamental solutionPauli, W., Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) {{ISBN|0486414620}} ; See section 44. $\backslash varphi(x,t)$ to

:$\backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; =\; \backslash frac\{\backslash partial^2\; \backslash varphi\}\{\backslash partial\; x^2\}-x^2\backslash varphi\; \backslash equiv\; D\_x\; \backslash varphi\; ~.$

The orthonormal eigenfunctions of the operator $D$ are the Hermite functions,

:$\backslash psi\_n\; =\; \backslash frac\{H\_n(x)\; \backslash exp(-x^2/2)\}\{\backslash sqrt\{2^n\; n!\; \backslash sqrt\{\backslash pi\}\}\},$

with corresponding eigenvalues $(-2n-1)$, furnishing particular solutions

: $\backslash varphi\_n(x,\; t)=\; e^\{-(2n+1)t\}\; ~H\_n(x)\; \backslash exp(-x^2/2)\; ~.$

The general solution is then a linear combination of these; when fitted to the initial condition $\backslash varphi(x,0)$, the general solution reduces to

: $\backslash varphi(x,t)=\; \backslash int\; K(x,y;t)\; \backslash varphi(y,0)\; dy\; ~,$

where the kernel $K$ has the separable representation

:$K(x,y;t)\backslash equiv\backslash sum\_\{n\backslash ge\; 0\}\; \backslash frac\; \{e^\{-(2n+1)t\}\}\{\backslash sqrt\backslash pi\; 2^n\; n!\}\; ~\; H\_n(x)\; H\_n(y)\; \backslash exp(-(x^2+y^2)/2)~.$

Utilizing Mehler's formula then yields

:$\{\backslash sum\_\{n\backslash ge\; 0\}\; \backslash frac\; \{(\backslash rho/2)^n\}\{n!\}\; H\_n(x)\; H\_n(y)\; \backslash exp(-(x^2+y^2)/2)\; =\; \{1\backslash over\; \backslash sqrt\{(1-\backslash rho^2)\}\}\; \backslash exp\backslash left(\{4xy\backslash rho\; -\; (1+\backslash rho^2)(x^2+y^2)\backslash over\; 2(1-\backslash rho^2)\}\backslash right)\}~.$

On substituting this in the expression for $K$ with the value $e^\{-2t\}$ for $\backslash rho$, Mehler's kernel finally reads

{{Equation box 1

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|equation = $K(x,y;t)=\; \backslash frac\{1\}\{\backslash sqrt\{2\backslash pi\backslash sinh(2t)\}\}~\backslash exp\backslash left(-\backslash coth(2t)~(x^2+y^2)/2\; +\; \backslash operatorname\{csch\}(2t)~xy\backslash right).$

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When $t\; =\; 0$, variables $x$ and $y$ coincide, resulting in the limiting formula necessary by the initial condition,

:$K(x,y;0)=\; \backslash delta(x-y)~.$

As a fundamental solution, the kernel is additive,

:$\backslash int\; dy\; K(x,y;t)\; K(y,z;t\text{'})\; =\; K(x,z;t+t\text{'})\; ~.$

This is further related to the symplectic rotation structure of the kernel $K$.The quadratic form in its exponent, up to a factor of −1/2, involves the simplest (unimodular, symmetric) symplectic matrix in Sp(2,R). That is,

:$(x,y)\; \{\backslash mathbf\; M\}\; \backslash begin\{pmatrix\}\; x\; \backslash \backslash \; y\; \backslash end\{pmatrix\}\; ~,~$   where

:

so it preserves the symplectic metric,

:

When using the usual physics conventions of defining the quantum harmonic oscillator instead via

:$i\; \backslash frac\{\backslash partial\; \backslash varphi\}\{\backslash partial\; t\}\; =\; \backslash frac\{1\}\{2\}\backslash left(-\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}+x^2\backslash right)\; \backslash varphi\; \backslash equiv\; H\; \backslash varphi,$

and assuming natural length and energy scales, then the Mehler kernel becomes the Feynman propagator $K\_\{H\}$ which reads

:

i.e. $K\_\{H\}(x,y;t)\; =\; K(x,y;\; i\; t/2\; ).$

When $t>\backslash pi$ the $i\; \backslash sin\; t$ in the inverse square-root should be replaced by $|\backslash sin\; t|$ and $K\_\{H\}$ should be multiplied by an extra Maslov phase factor {{Cite journal|

journal=International Journal of Theoretical Physics|

date=1979 |

volume =18 |

issue = 4 |

page= 245-250|

title= Extended Feynman Formula for Harmonic Oscillator|

author1= Horvathy, Peter

|

doi=10.1007/BF00671761 |

bibcode=1979IJTP...18..245H |

s2cid=117363885 }}

:$\backslash exp\backslash left(i\backslash theta\_\{\backslash rm\; Maslov\}\backslash right)\; =\; \backslash exp\backslash left(-i\backslash frac\{\; \backslash pi\}\{\; 2\}\backslash left(\backslash frac\; \{1\}\{2\}\; +\backslash left\backslash lfloor\backslash frac\{t\}\{\backslash pi\}\backslash right\backslash rfloor\; \backslash right)\backslash right).$

When $t\; =\; \backslash pi/2$ the general solution is proportional to the Fourier transform $\backslash mathcal\{F\}$ of the initial conditions $\backslash varphi\_0(y)\backslash equiv\backslash varphi(y,0)$ since

: $\backslash varphi(x,\; t=\backslash pi/2)\; =\; \backslash int\; K\_\{H\}(x,y;\; \backslash pi/2)\; \backslash varphi(y,0)\; dy\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\; i\}\}\; \backslash int\; \backslash exp(-i\; x\; y)\; \backslash varphi(y,0)\; dy\; =\; \backslash exp(-i\; \backslash pi\; /4)\; \backslash mathcal\{F\}[\backslash varphi\_0](x)\; ~,$

and the exact Fourier transform is thus obtained from the quantum harmonic oscillator's number operator written as{{citation |last1=Wolf |first1=Kurt B. |title=Integral Transforms in Science and Engineering |year=1979 |publisher=Springer}} ([https://doi.org/10.1007/978-1-4757-0872-1] and [https://www.fis.unam.mx/~bwolf/integraleng.html]); see section 7.5.10.

:$N\; \backslash equiv\; \backslash frac\{1\}\{2\}\backslash left(x-\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash right)\backslash left(x+\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash right)\; =\; H-\backslash frac\{1\}\{2\}\; =\; \backslash frac\{1\}\{2\}\backslash left(-\backslash frac\{\backslash partial^2\}\{\backslash partial\; x^2\}+x^2-1\backslash right)\; ~$

since the resulting kernel

:$\backslash langle\; x\; \backslash mid\; \backslash exp\; (-it\; N)\; \backslash mid\; y\; \backslash rangle\; \backslash equiv\; K\_\{N\}(x,y;t)\; =\; \backslash exp(i\; t\; /2)\; K\_\{H\}(x,y;\; t)\; =\; \backslash exp(i\; t\; /2)\; K(x,y;i\; t\; /2)$

also compensates for the phase factor still arising in $K\_\{H\}$ and $K$, i.e.

:$\backslash varphi(x,t\; =\; \backslash pi/2)=\; \backslash int\; K\_\{N\}(x,y;\; \backslash pi/2)\; \backslash varphi(y,0)\; dy\; =\; \backslash mathcal\{F\}[\backslash varphi\_0](x)~,$

which shows that the number operator can be interpreted via the Mehler kernel as the generator of fractional Fourier transforms for arbitrary values of $t$, and of the conventional Fourier transform $\backslash mathcal\{F\}$ for the particular value $t\; =\; \backslash pi/2$, with the Mehler kernel providing an active transform, while the corresponding passive transform is already embedded in the basis change from position to momentum space. The eigenfunctions of $N$ are the usual Hermite functions $\backslash psi\_n(x)$ which are therefore also Eigenfunctions of $\backslash mathcal\{F\}$.{{Cite journal|

journal=Symmetry |

date=2021 |

volume =13 |

issue = 5 |

title=Hermite Functions and Fourier Series |

author1= Celeghini, Enrico |

author2= Gadella, Manuel |

author3= del Olmo, Mariano A. |

page=853 |

doi = 10.3390/sym13050853 |

arxiv=2007.10406 |

bibcode=2021Symm...13..853C |

doi-access=free }}

There are many proofs of the formula.

The formula is a special case of the Hardy–Hille formula, using the fact that the Hermite polynomials are a special case of the associated Laguerre polynomials:$$\backslash begin\{align\}$$

H_{2n}(x) &= (-1)^n 2^{2n} n! L_n^{(-1/2)} (x^2) \\[4pt]

H_{2n+1}(x) &= (-1)^n 2^{2n+1} n! x L_n^{(1/2)} (x^2)

\end{align}The formula is a special case of the Kibble–Slepian formula, so any proof of it immediately yields of proof of the Mehler formula.{{Cite journal |last=Ismail |first=Mourad E. H. |last2=Zhang |first2=Ruiming |date=2017-04-01 |title=A review of multivariate orthogonal polynomials |url=https://www.sciencedirect.com/science/article/pii/S1110256X16300761#bib0028 |journal=Journal of the Egyptian Mathematical Society |volume=25 |issue=2 |pages=91–110 |doi=10.1016/j.joems.2016.11.001 |issn=1110-256X |doi-access=free}}

Foata gave a combinatorial proof of the formula.{{Cite journal |last=Foata |first=Dominique |date=1978-05-01 |title=A combinatorial proof of the Mehler formula |url=https://www.sciencedirect.com/science/article/pii/0097316578900663 |journal=Journal of Combinatorial Theory, Series A |volume=24 |issue=3 |pages=367–376 |doi=10.1016/0097-3165(78)90066-3 |issn=0097-3165|url-access=subscription }}

Hardy gave a simple proof by the Fourier integral representation of Hermite polynomials.{{Cite journal |last=Watson |first=G. N. |date=July 1933 |title=Notes on Generating Functions of Polynomials: (2) Hermite Polynomials |url=http://doi.wiley.com/10.1112/jlms/s1-8.3.194 |journal=Journal of the London Mathematical Society |language=en |volume=s1-8 |issue=3 |pages=194–199 |doi=10.1112/jlms/s1-8.3.194|url-access=subscription }} Using the Fourier transform of the gaussian $e^\{-x^2\}=\backslash frac\{1\}\{\backslash sqrt\{\; \backslash pi\}\}\; \backslash int\; e^\{-t^2+2\; i\; x\; t\}\; dt$, we have$$H\_n(x)\; =\; (-1)^n\; e^\{x^2\}\; \backslash frac\; \{d^n\}\{dx^n\}\; e^\{-x^2\}\; =\; \backslash frac\{e^\{x^2\}\}\{\backslash sqrt\{\backslash pi\}\}\; \backslash int\; (-2it)^n\; e^\{-t^2+2\; i\; x\; t\}\; d\; t$$from which the summation $\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{(\backslash rho/2)^n\}\{n!\}\; \backslash mathit\{H\}\_n(x)\backslash mathit\{H\}\_n(y)$ converts to a double integral over a summation$$\backslash frac\{e^\{x^2+y^2\}\}\{\backslash pi\}\; \backslash iint\_\{\backslash mathbb\{R\}^2\}\; e^\{-\backslash left(t^2+s^2\backslash right)+2\; i\; x\; t+2\; i\; y\; s\}\; \backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{\backslash left(-\; 2\; t\; s\backslash rho\backslash right)^n\}\{n!\}d\; t\; d\; s$$which can be evaluated directly as two gaussian integrals.

The result of Mehler can also be linked to probability. For this, the variables should be rescaled as $x\; \backslash to\; x/\backslash sqrt\{2\}$, $y\; \backslash to\; y/\backslash sqrt\{2\}$, so as to change from the "physicist's" Hermite polynomials $H(\backslash cdot)$ (with weight function $\backslash exp(-x^2)$) to "probabilist's" Hermite polynomials $\backslash operatorname\{He\}(\backslash cdot)$ (with weight function $\backslash exp(-x^2/2)$). They satisfy$$H\_n(x)=2^\backslash frac\{n\}\{2\}\; \backslash operatorname\{He\}\_n\backslash left(\backslash sqrt\{2\}\; \backslash ,x\backslash right),\; \backslash quad\; \backslash operatorname\{He\}\_n(x)=2^\{-\backslash frac\{n\}\{2\}\}\; H\_n\backslash left(\backslash frac\; \{x\}\{\backslash sqrt\; 2\}\; \backslash right).$$Then, $E$ becomes

:$$

\frac 1{\sqrt{1-\rho^2}}\exp\left(-\frac{\rho^2 (x^2+y^2)- 2\rho xy}{2(1-\rho^2)}\right)

= \sum_{n=0}^\infty \frac{\rho^n}{n!} ~ \operatorname{He}_n(x)\operatorname{He}_n(y) ~.

The left-hand side here is $p(x,y)/p(x)p(y)$ where $p(x,y)$ is the bivariate Gaussian probability density function for variables $x,y$ having zero means and unit variances:

:$p(x,y)\; =$

\frac 1{2\pi \sqrt{1-\rho^2}}\exp\left(-\frac{(x^2+y^2)- 2\rho xy}{2(1-\rho^2)}\right) ~,

and $p(x),\; p(y)$ are the corresponding probability densities of $x$ and $y$ (both standard normal).

There follows the usually quoted form of the result (Kibble 1945){{Cite journal

| last1=Kibble | first1=W. F.

| title=An extension of a theorem of Mehler's on Hermite polynomials

| doi=10.1017/S0305004100022313

| mr=0012728

| date=1945

| journal=Mathematical Proceedings of the Cambridge Philosophical Society

| volume=41

| issue=1

| pages=12–15| bibcode=1945PCPS...41...12K | s2cid=121931906}}

:$p(x,y)\; =\; p(x)\; p(y)\backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\{\backslash rho^n\}\{n!\}\; ~\; \backslash operatorname\{He\}\_n(x)\backslash operatorname\{He\}\_n(y)\; ~.$

The exponent can be written in a more symmetric form:$$$$

\frac 1{\sqrt{1-\rho^2}}\exp\left(\frac{\rho(x+y)^2}{4(1+\rho)}-\frac{\rho(x-y)^2}{4(1-\rho)}\right)

= \sum_{n=0}^\infty \frac{\rho^n}{n!} ~ \operatorname{He}_n(x)\operatorname{He}_n(y) ~.This expansion is most easily derived by using the two-dimensional Fourier transform of $p(x,y)$, which is

:$c(iu\_1,\; iu\_2)\; =\; \backslash exp\; (-\; (u\_1^2\; +\; u\_2^2\; -\; 2\; \backslash rho\; u\_1\; u\_2)/2)~.$

This may be expanded as

:$\backslash exp(\; -(u\_1^2\; +\; u\_2^2)/2\; )\; \backslash sum\_\{n=0\}^\backslash infty\; \backslash frac\; \{\backslash rho^n\}\{n!\}\; (u\_1\; u\_2)^n\; ~.$

The Inverse Fourier transform then immediately yields the above expansion formula.

This result can be extended to the multidimensional case.{{Citation | last1=Slepian | first1=David | title=On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials | doi=10.1137/0503060 | mr=0315173 | year=1972 | journal=SIAM Journal on Mathematical Analysis | issn=0036-1410 | volume=3 | issue=4 | pages=606–616}}{{Cite journal|

journal = Mathematische Zeitschrift |

date=1995 |

volume =219 |

pages=413–449 |

title=Symplectic classification of quadratic forms, and general Mehler formulas |

author= Hörmander, Lars | doi = 10.1007/BF02572374|

s2cid=122233884 }}

Erdélyi gave this as an integral over the complex plane{{Cite journal |last=Erdélyi |first=Artur |date=1939-12-01 |title=Über eine erzeugende Funktion von Produkten Hermitescher Polynome |url=https://link.springer.com/article/10.1007/BF01210650 |journal=Mathematische Zeitschrift |language=de |volume=44 |issue=1 |pages=201–211 |doi=10.1007/BF01210650 |issn=1432-1823|url-access=subscription }}$$\backslash sum\_\{n=0\}^\{\backslash infty\}\; \backslash frac\{\backslash rho^n\}\{n!\}\; \backslash operatorname\{He\}\_n(x)\; \backslash operatorname\{He\}\_n(y)$$

=\frac{1}{\pi t} \iint \exp \left[-\frac{u^2+v^2}{\rho}+(u+i v) x+(u-i v) y-\frac{1}{2}(u+i v)^2-\frac{1}{2}(u-i v)^2\right] d u d v .which can be integrated with two gaussian integrals, yielding the Mehler formula.

{{main|Fractional Fourier transform}}

Since Hermite functions $\backslash psi\_n$ are orthonormal eigenfunctions of the Fourier transform,

:$\backslash mathcal\{F\}\; [\backslash psi\_n](y)=(-i)^n\; \backslash psi\_n(y)\; ~,$

in harmonic analysis and signal processing, they diagonalize the Fourier operator,

:$\backslash mathcal\{F\}[f](y)\; =\backslash int\; dx\; f(x)\; \backslash sum\_\{n\backslash geq\; 0\}\; (-i)^n\; \backslash psi\_n(x)\; \backslash psi\_n(y)\; ~.$

Thus, the continuous generalization for real angle $\backslash alpha$ can be readily defined (Wiener, 1929;Wiener, N (1929), "Hermitian Polynomials and Fourier Analysis", Journal of Mathematics and Physics 8: 70–73. Condon, 1937Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA 23, 158–164. [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1076889/pdf/pnas01779-0028.pdf online]), the fractional Fourier transform (FrFT), with kernel

:$\backslash mathcal\{F\}\_\backslash alpha\; =\; \backslash sum\_\{n\backslash geq\; 0\}\; (-i)^\{2\backslash alpha\; n/\backslash pi\}\; \backslash psi\_n(x)\; \backslash psi\_n(y)\; ~.$

This is a continuous family of linear transforms generalizing the Fourier transform, such that, for $\backslash alpha\; =\; \backslash pi/2$, it reduces to the standard Fourier transform, and for $\backslash alpha\; =\; -\backslash pi/2$ to the inverse Fourier transform.

The Mehler formula, for $\backslash rho\; =\; \backslash exp(-i\backslash alpha)$, thus directly provides

:$\backslash mathcal\{F\}\_\backslash alpha[f](y)\; =$

\sqrt{\frac{1-i\cot(\alpha)}{2\pi}} ~ e^{i \frac{\cot(\alpha)}{2} y^2}

\int_{-\infty}^\infty

e^{-i\left(\csc(\alpha)~ y x - \frac{\cot(\alpha)}{2} x^2\right )} f(x)\, \mathrm{d}x ~.

The square root is defined such that the argument of the result lies in the interval $[-\backslash pi/2,\; \backslash pi/2]$.

If $\backslash alpha$ is an integer multiple of $\backslash pi$, then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, $\backslash delta(x-y)$ or $\backslash delta(x+y)$, for $\backslash alpha$ an even or odd multiple of $\backslash pi$, respectively. Since $\backslash mathcal\{F\}^2[f]\; =\; f(-x)$, $\backslash mathcal\{F\}\_\backslash alpha[f]$ must be simply $f(x)$ or $f(-x)$ for $\backslash alpha$ an even or odd multiple of $\backslash pi$, respectively.

• {{slink|Oscillator representation#Harmonic oscillator and Hermite functions}}
• Heat kernel
• Hermite polynomials
• Parabolic cylinder functions
• {{slink|Laguerre polynomials#Hardy–Hille formula}}

• Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). Heat Kernels and Dirac Operators, (Springer: Grundlehren Text Editions) Paperback {{ISBN|3540200622}}
• {{Cite journal

| last1=Louck | first1=J. D.

| journal=Advances in Applied Mathematics

| volume=2

| date= 1981

| pages= 239–249

| title=Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods

| issue=3

| doi=10.1016/0196-8858(81)90005-1| doi-access=free}}

• {{cite journal

| last1=Srivastava | first1=H. M.

| last2=Singhal | first2=J. P.

| date=1972

| title=Some extensions of the Mehler formula

| journal=Proceedings of the American Mathematical Society

| volume=31

| pages=135–141

| doi=10.1090/S0002-9939-1972-0285738-4 | doi-access=free}}

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